Lengths spectrum of hyperelliptic components

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmüller geodesic flow restricted to the hyperelliptic connected components of the strata of Abelian differentials, settling a question of [Far06]; This is the first description of the systole in any component of any stratum outside of genus two and three. We stress that all proofs and computations can be performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components and we provide a picture of that for small genera.